samplemidterm2solutions

samplemidterm2solutions - Sample Midterm 2 Math 33A Fall...

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Unformatted text preview: Sample Midterm 2, Math 33A, Fall 2007 November 19, 2007 1. (2 pts each) Indicate whether each statement is true or false; you need not show your work here. If T : R n → R n is an invertible linear transformation, B is a basis of R n , and B is the matrix of T with respect to the basis B , then B must be invertible. T F If A is a n by n matrix with exactly one 1 in each row, exactly one 1 in each column, and 0s elsewhere, then det A must be 1 or- 1. T F If A is a n by n matrix, then det 2 A must be equal to 2(det A ). T F If ~v is a vector in R n and W is a subspace of R n , then the length of proj W ~v must be less than or equal to the length of ~v . T F . If W and V are subspaces of R n and W ⊂ V , then it must also be true that W ⊥ ⊂ V ⊥ . T F . Solution. T, T, F, T, F. If T : R n → R n is an invertible linear transformation and A is its matrix with respect to the standard basis, then A is invertible. If S is the change-of-basis matrix for the basis B , then B = S- 1 AS , which is a product of invertible matrices. Hence, which is a product of invertible matrices....
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samplemidterm2solutions - Sample Midterm 2 Math 33A Fall...

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